It is a direct consequence of Abhyankar's Conjecture (which was proved by Raynaud and Harbater) that if $k$ is an algebraically closed field of characteristic $p > 0$, then no affine curve $X_{/k}$ has trivial etale fundamental group. (Note that for curves, affine = quasi-projective, non-projective, by Riemann-Roch.)

I have some lecture notes on this subject from years back:

http://alpha.math.uga.edu/~pete/fundincharp.pdf

(In contrast to what it says on the first page, they are from 2002.)

**Addendum**: The comments above give plenty of examples of non-projective quasi-projective varieties with trivial etale fundamental group in characteristic $p$ (or really in characteristic *quelconque*). An interesting question left open by these examples is whether there are any (nontrivial) simply connected **affine** varieties in characteristic $p$. As I have said, the answer is "no" in dimension one.